Comment on “On Quantum Mechanical Operators in Generalized Coordinates”

Gary R. Gruber

Gary R. Gruber

Utpal Roy, Suranjana Ghosh, T. Shreecharan +1 more

In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in spherical polar coordinate comes. We have tried to point out most of the elementary quantum mechanical results, related to the momentum operator, which has coordinate dependence. We explicitly calculate the momentum expectation values in various bound states and show that the expectation value really turns out to be zero, a consequence of the fact that the momentum expectation value is real. We comment briefly on the status of the angular variables in quantum mechanics and the problems related in interpreting them as dynamical variables. At the end, we calculate the Heisenberg's equation of motion for the radial component of the momentum for the Hydrogen atom.

Makoto Kato

R. Bonifacio, Lorenzo M. Narducci, E. Montaldi

Stanley A. Bruce, A. Delgado, Luis Roa

We study minimal interactions in quantum systems characterized by position and momentum operators defined as the direct product of a traceless finite matrix and an ordinary canonical coordinate.